GCSE · Maths · AQA · Spec 8300

Proportion as equality of ratios

Double a paint mix and you get the same colour. Add a tin of each colour and you don't. Why does multiplying work when adding fails?

Ratio · Proportion

More paint, same green

Batches mixed: 1/6. Blue : yellow (litres) 2 : 3. Both parts multiplied by × 1. Simplest form 2 : 3. Yellow minus blue 1 litre

Blue : yellow (litres)2 : 3Both parts multiplied by× 1Simplest form2 : 3Yellow minus blue1 litre

Each cell of the bar is one batch of green paint: 2 litres of blue and 3 litres of yellow. Drag to mix more batches. Watch which readouts change, and which one never does.

Exam line: Two ratios are in proportion when they are equal: one is the other with both parts multiplied by the same number, so both simplify to the same ratio.

Which idea do you hold?

Same green or not?

A tin of green is mixed from blue and yellow in the ratio 4 : 6. Two friends try to make more of the same green. Ali uses 6 blue to 8 yellow. Bea uses 6 blue to 9 yellow.

Which is closest to what you think right now?

Ali's 6 : 8 matches, because he added 2 to each colour, so nothing changed.Bea's 6 : 9 matches, because 4 : 6 and 6 : 9 both simplify to 2 : 3.Neither matches. You can't get from 4 to 6 by multiplying by a whole number.Both match. Each one still has more yellow than blue.

How sure are you?

I'm sureI think soI'm guessing

Ali's 6 : 8 matches, because he added 2 to each colour, so nothing changed.

This is a common misconception

Adding 2 to each part kept the gap at 2, but the gap isn't what matters. Simplify both: 4 : 6 is 2 : 3, while 6 : 8 is 3 : 4. Different ratios, so a different green. In 4 : 6 there's one and a half times as much yellow as blue; in 6 : 8 there's only one and a third times as much, so Ali's mix is bluer.

Bea's 6 : 9 matches, because 4 : 6 and 6 : 9 both simplify to 2 : 3.

This matches the evidence

Yes. 4 : 6 divided by 2 is 2 : 3, and 6 : 9 divided by 3 is 2 : 3. Same simplest form, so the ratios are equal and the mixes are in proportion. You can also see it as one multiplier: 4 × 1.5 = 6 and 6 × 1.5 = 9.

Neither matches. You can't get from 4 to 6 by multiplying by a whole number.

This is a common misconception

The multiplier doesn't have to be a whole number. Bea multiplied both parts by 1.5: 4 × 1.5 = 6 and 6 × 1.5 = 9. When the multiplier is awkward, simplify both ratios instead: 4 : 6 and 6 : 9 both come out as 2 : 3.

Both match. Each one still has more yellow than blue.

This is a common misconception

Having more yellow isn't enough; it's how much more, compared with the blue, that sets the colour. 6 : 8 simplifies to 3 : 4 and 6 : 9 simplifies to 2 : 3. Only one of those is the original 2 : 3.

Two ratios are equal when one multiplier takes both parts of the first to both parts of the second, or, the same test, when they simplify to the same ratio. Adding the same amount to both parts is not the test.

Quick sort

Same green or a different green?

Is each mix in proportion with 2 : 3?

Still to sort

Same green: equal to 2 : 3 (0)

One multiplier takes 2 and 3 to both new numbers.

Where the line is: The test is one multiplier for both parts, or the same simplest form. A mix that grows, or still has more yellow, can still fail it.

Different green: not equal to 2 : 3 (0)

No single multiplier works on both parts.

6 of 6 still to sort.

Every card is a new mix of blue : yellow. Put each one where it belongs, then read why.

Your turn to use it

Find the missing amount

A builder mixes cement and sand in the ratio 3 : 5. She uses 12 bags of cement. How many bags of sand does she need to keep the same mix?

cement : sand = 3 : 5Start with the ratio you know.
3 : 5 = 12 : xThe new mix must be an equal ratio. Keep the order the same on both sides: cement first, sand second. x is the bags of sand.
missing step

Which line is step 3?

12 − 3 = 9, so add 9 to both parts12 ÷ 3 = 4, so both parts are multiplied by 412 ÷ 5 = 2.4, so both parts are multiplied by 2.4

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

2 : 3 = 4 : 6 = 6 : 9

Different numbers, the same ratio. When two ratios are equal like this, they are in proportion.

What you need to know

Two ratios are in proportion when they are equal.
You get an equal ratio by multiplying, or dividing, both parts by the same number. That number doesn't have to be whole.
To test two ratios, simplify both: equal ratios have the same simplest form.
To find a missing value, write the two ratios equal in the same order, find the multiplier, then use it on the other part.

The big picture

Two ratios are in proportion when they are equal: one is the other with both parts multiplied (or divided) by the same number, so both simplify to the same ratio. You can test a pair by simplifying both, and you can use the equality to find a missing value: set the ratios equal in the same order, find the multiplier, and apply it to the other part.

Key points

1'In proportion' means 'equal ratios': 2 : 3 = 4 : 6 = 6 : 9.

2Equal ratios share a multiplier, not a difference: 2 : 3 → 4 : 6 doubles both parts, while the gap grows from 1 to 2.

3Simplest form is the quickest test: 4 : 6 and 6 : 9 both simplify to 2 : 3.

4Missing value: match the parts in the same order, find the multiplier from a known pair, apply it to the unknown's partner, then check by simplifying.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

A garden mix uses compost and soil in the ratio 6 : 4. How many buckets of compost go with 10 buckets of soil?

⚠ Watch out

Adding the same amount to both parts doesn't give an equal ratio: 3 : 4 → 5 : 6 is a different mix. Multiply both parts by the same number instead.

🧠

Memory hook

Same recipe, bigger batch: multiply every part by the same number. Multiply, don't add.

✓

Check yourself

Are 5 : 3 and 20 : 12 in proportion? Say which multiplier links them, then check by simplifying 20 : 12.

Flashcards

(6)

What does it mean for two ratios to be 'in proportion'?

The two ratios are equal, like 2 : 3 and 8 : 12.

Between two equal ratios, what stays the same: the multiplier or the difference?

The multiplier. Both parts are multiplied by the same number, while the gap between the parts can change.

Quickest way to test whether two ratios are equal?

Simplify both. If they reduce to the same simplest form, they're equal.

Does the multiplier between two equal ratios have to be a whole number?

No. 4 : 6 → 6 : 9 multiplies both parts by 1.5.

In a : b = c : x, where does the multiplier come from?

From the pair you know on both sides, a → c. Then multiply b by it to get x.

Why must both ratios be written in the same order?

Each part has to line up with its partner (cement with cement, sand with sand). Swap the order and you compare the wrong amounts.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Same green or not?

Same green or a different green?

Find the missing amount

More AQA GCSE Maths topics